Domanda del libro di testo
Repeating Decimals
Express each of the numbers in Exercises 23–30 as the ratio of two integers.
3.1̅4̅2̅8̅5̅7 = 3.142857142857 ...
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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 10, Problema 10.2.64
Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
∑ (from n = 1 to ∞) (1 − 1/n)ⁿ
Guida verificata passo dopo passo1
First, identify the general term of the series: \(a_n = \left(1 - \frac{1}{n}\right)^n\).
Next, analyze the behavior of the term \(a_n\) as \(n\) approaches infinity by finding the limit \(\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n\).
Recall the known limit \(\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n = e^{-1}\), which is a nonzero constant.
Since the terms \(a_n\) do not approach zero, apply the Divergence Test (also called the nth-term test for divergence), which states that if \(\lim_{n \to \infty} a_n \neq 0\), then the series \(\sum a_n\) diverges.
Conclude that the series \(\sum_{n=1}^\infty \left(1 - \frac{1}{n}\right)^n\) diverges because its terms do not tend to zero, so it does not converge and therefore does not have a finite sum.
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Convergence and Divergence of Infinite Series
An infinite series converges if the sequence of its partial sums approaches a finite limit; otherwise, it diverges. Determining convergence involves testing whether the terms decrease sufficiently fast and if the sum stabilizes as more terms are added.
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Percorso guidato
06:52
Convergence of an Infinite Series
Limit of the General Term
A necessary condition for series convergence is that the general term approaches zero as n approaches infinity. If the limit of the term (1 - 1/n)^n is not zero, the series must diverge by the Test for Divergence.
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05:50One-Sided Limits
Behavior of the Term (1 - 1/n)^n
The expression (1 - 1/n)^n approaches 1/e as n becomes large. Understanding this limit helps determine the behavior of the series terms and whether the series can converge or diverge.
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Percorso guidato
05:44Divergence Test (nth Term Test)
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Domanda del libro di testo
Finding a Sequence’s Formula
In Exercises 13–30, find a formula for the nth term of the sequence.
1, -4, 9, -16, 25, …Squares of the positive integers, with alternating signs
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Domanda del libro di testo
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ ( n / (n + 1) )ⁿ^ ² ] xⁿ (Hint: Apply the Root Test.)
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Domanda del libro di testo
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?
∑ (from n = 0 to ∞) (x + 5)ⁿ
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Domanda del libro di testo
Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = x³ − 2x + 4,a = 2
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Domanda del libro di testo
Is it true that a sequence {aₙ} of positive numbers must converge if it is bounded above? Give reasons for your answer.
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